Mathematical Models of Uptake, Metabolism & Response Likelihood

Exponent’s environmental scientists have increasingly turned to mathematical modeling techniques to provide broad information about environmental contaminant levels and assess the potential variability in contaminant levels caused by environmental factors. The air, water, soil, and biota of the earth form a complex and interacting environment that challenges environmental assessors. While measurements of environmental contaminants are preferred, it is often impractical or impossible to measure levels in the full geographic scope of interest or during the full range of possible environmental conditions (e.g., differences in stream flow, meteorology, etc.). Furthermore, in some cases, the object causing the potential environmental damage does not yet exist (e.g., a new industrial facility or new land use project).

Chemical Fate and Transport

Chemicals released to the environment can change their form, such as the rusting of iron, and can cross into other media, such as mercury being released into the air and eventually becoming methyl mercury when it interacts with water and eventually finds its way into some fish populations.. Understanding the processes by which chemicals change and travel in the environment is key to assessing potential chemical exposure. An intimate understanding of chemical fate and transport provides the basis for identifying the potential exposure pathways and quantifying the magnitude, frequency, and duration of exposure. The evaluation of chemical fate and transport is a critical step in risk assessment, exposure characterization, and dose reconstruction.

Exponent scientists and engineers provide the full spectrum of expertise needed to evaluate chemical fate and transport. For air pollutants, our knowledge and experience can characterize the processes by which the pollutants are formed and the magnitude and rate of emissions from the sources, followed by using the appropriate modeling tools and meteorological and geographical data to quantify the potential magnitude of exposure. We have evaluated pollutants released into fresh or marine waters, using our broad understanding of the hydrodynamic and sediment transport processes involved, the geochemical variables of chemical transport and transformation, and the biological and ecological factors involved in direct-contact exposure or indirect exposure through the food chain. Our scientists and engineers have also evaluated pollutants in soil, whether deposited directly by unintentional spills or indirectly through atmospheric deposition or floods, with a full understanding of the complex variables involved in exposure.

Air Dispersion Modeling

Atmospheric dispersion modeling is a widely used methodology for estimating ambient air concentrations of pollutants, based on estimates of pollutant emissions and a characterization of meteorological conditions in the area around the source.

Exponent has expertise with all of the most commonly applied dispersion models, including AERMOD, ISCST3, CALPUFF, BLP, SLAB, and ALOHA. Additionally, our scientists have developed the Probabilistic Exposure and Risk model for FUMigants (PERFUM), a dispersion and risk assessment model used for evaluating the risk to bystanders associated with fumigant chemicals. PERFUM is used by EPA for regulatory decision making regarding fumigants.

Exponent uses dispersion modeling to assist clients in addressing a variety of scientific issues:

  • Evaluation of air pollution impacts from existing emission sources including industrial facilities (e.g., chemical, petroleum, or utility sectors), agricultural sources, roads, construction projects, contaminated waste sites, and others 
  • Development of a prospective tool to evaluate the potential impact (or benefit) of a planned project (e.g., new facility, addition to an existing facility, installation of emission controls, etc.) 
  • Analysis of an accidental release of a gas (e.g., a tank explosion or highway accident) 
  • Assessment of historical sources (i.e., sources that no longer exist) that may be the subject of litigation 

Concentration estimates from air dispersion models can be used in risk assessment to estimate potential health risks to affected populations. Exponent has expertise in both air dispersion modeling and risk assessment.


Water-borne contaminants evaluated by Exponent include chemical and biological agents (pathogens). Exposures to water-born contaminants can occur in a number of scenarios, including:

  • Inhalation of volatile organic chemicals in the groundwater volatilized into buildings – vapor intrusion 
  • Inhalation and dermal contact to chemicals during bathing or showering 
  • Drinking water contaminated with chemicals or pathogens 
  • Dermal and/or incidental ingestion of contaminants in the water at beaches, rivers, or lakes during recreational activities 

Chemicals that are hydrophobic tend to bind to suspended particulates in water and require special consideration. Hydrophobic chemicals can bind to sediments in a riverine, lake, or ocean setting and biomagnify through the food chain. As a result, the consumption of fish and shellfish contaminated with chemicals such as dioxin, mercury, and PCBs through water, sediments, and the food chain can become a major concern.

Exposure modeling of water-borne contaminants involves an understanding of their physicochemical properties (or biological properties for pathogens) and environmental fate and transport. Information on behavioral factors, such as the water contact rate of the exposed population, is key to accurately modeling real-world exposures.

Exponent engineers and scientists understand the important issues involved with modeling exposures to water-borne contaminants. Our staff has modeled a wide range of exposure scenarios to characterize exposures for general consideration (e.g., environmental standard setting) and for site-specific situations. We also have expertise using a number of off-the-shelf models required by regulatory agencies.

Metabolism & Response Likelihood

Exponent has extensive capabilities for and experience developing and applying mathematical models that address exposure-pathway-specific uptake, distribution, metabolism, and risk of toxic response for chemicals encountered in the workplace and through environmental exposure scenarios. For example, Exponent recently developed rat and human physiological based pharmacokinetic (PBPK) models for malathion, an organophosphate pesticide that has been applied worldwide for many decades. Figure 1 shows how this model predicts the cumulative urinary excretion of four major metabolites in rats exposed to malathion over a three-day period.

Mathematical Exposure Modeling

Figure 1. PBPK model prediction of major malathion metabolites (DCA, MCA, DMT, and DMP) excreted in urine after adult rats are exposed by gavage on three consecutive days to 69 mg/day of malathion (top left) , and peak inhibition of red blood cell (RBC) acetylcholinesterase (AChE) after daily or chronic administration of malathion to rat pups and adults (top right), compared to experimental data (points). (Bottom) PBPK-model predictions of AChE inhibition (left) and blood malathion concentration (right) after 0.025-kg (black curve) or 0.25 kg (red curve) rats ingest 10.5 or 7.7 mg/kg/day, respectively, over an 8-hour period on each day.

Mathematical Modeling Publications 

Bogen, KT, Heilman J. Reassessment of MTBE cancer potency considering modes of action for MTBE and its metabolites. Crit Rev Toxicol 2015 doi.10.3109/10408444.2015.1052367 (in press).

Bogen KT, Sheehan PS. Dermal vs. total uptake of benzene from mineral spirits solvent during parts washing. Risk Anal 2014; 34(7):1336–1358.

Bogen KT, Reiss R. Generalized Haber's law for exponential concentration decline, with application to riparian-aquatic pesticide ecotoxicity. Risk Anal 2012; 32(2):250–258.

Bogen KT, Cullen AC, Frey HC, Price PS. Probabilistic exposure analysis for chemical risk characterization. Toxicol Sci 2009; 109(1):4–17.

KT Bogen and FJ Gouveia. 2008. Impact of spatiotemporal fluctuations in airborne chemical concentration on toxic hazard assessment. Journal of Hazardous Materials. 152. p.228-240. 

Reiss R, Anderson EL, Cross CE, Hidy G, Hoel D, McClellan R, Moolgavkar S. Evidence of health impacts of sulfate and nitrate containing particles in ambient air. Inhalat Toxicol 2007; 19:419–449.

Reiss R. Temporal trends and weekend–weekday differences for benzene and 1,3-butadiene in Houston, Texas. Atmos Environ 2006; 40:4711–4724. 

Reiss R, Griffin J. A probabilistic model for acute bystander exposure and risk assessment for soil fumigants. Atmos Environ 2006; 40:3548–3560. 

Reiss R, Ryan PB, Koutrakis P, Tibbetts S. Ozone reactive chemistry on interior latex paint. Environ Sci Technol 1995; 29:1906–1912. 

Reiss R, Ryan PB, Tibbetts S, Koutrakis P. Measurement of organic acids, aldehydes, and ketones in residential environments and their relation to ozone. J Air Waste Manage Assoc 1995; 45:811–822. 

Reiss R, Ryan PB, Koutrakis P. Modeling ozone deposition onto indoor residential surfaces, Environ Sci Technol 1994; 28:504–513.

Sheehan PJ, Warmerdam JM, Ogle S, Humphrey DN, Patenaude SM. Evaluating the risk to aquatic ecosystems posed by leachate from tire shred fill in roads using toxicity tests, toxicity identification evaluations, and groundwater modeling. Environ Toxicol Chem 2006; 25(2):400-411.